Discrete imperfect repair.

Abstract

A discrete imperfect repair model is studied, it consists of units that try to perform jobs dynamically in time and that at failure may be repaired and are allowed to try to do again the jobs in which they failed until either eventual completions of the corresponding jobs or unsuccessful repairs of failed units. Of interest then is the numbers of completed jobs by the units before the occurrences of unsuccessful repairs, that is, the waiting times for an unsuccessful repair in the model. In many practical situations some information about the model is provided in the form of parameters. These parameters are usually quantities that indicate different levels of difficulty of the units in performing the jobs and the chances of repairing successfully failed units at any particular jobs time. In this work we first characterize, by means of failure rates, the distribution of the waiting times for an unsuccessful repair in terms of those parameters. Then we find necessary conditions on the parameters of two models that yield stochastical improvements of the numbers of completed jobs of the units of one model with respect to the other model. Using those results some ageing properties for the waiting times for an unsuccessful repair can be inferred from the parameters of the model. A class of discrete multivariate increasing failure rate distributions is also introduced and discussed. Some particular cases of the model are studied in some detail, and examples, counterexamples, and illustrations are given throughout the work.